Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family rational maps, parametrized by step size. When these maps are Poisson respect quadratic structure we say that Γ has Kahan-Poisson property. We show if is connected, it property and only cloning vertices $1,2,\dots ,n$ , an arc i → j precisely when < j, all arcs having same value. also prove similar result for augmented graphs, which correspond deformed systems obtained their discretizations superintegrable as well Liouville integrable.